多线性分数次积分与Lipschitz函数生成的交换子的有界性

Abstract

Abstract:

Let $m\in{\Bbb N}$, $\vec{b}=(b_{1},\cdots ,b_{m})$ whose components are of locally integrable functions, and $\vec{f}=(f_{1},\cdots ,f_{m})$ where $f_{1},\cdots ,f_{m}\in L_{c}^{\infty}({\mathbf{R}}^{n}).$ Let $x\notin\bigcap\limits_{i=1}^{m}\mbox{supp}f_{i},$ then the commutator generated by the multilinear fractional integral is given by
$$I_{\alpha,m}^{\vec{b}}(\vec{f})(x) =\dint_{({\mathbf{R}}^{n})^{m}}K(x,y_{1},\cdots ,y_{m})\prod\limits_{i=1}^{m}(b_{i}(x)-b_{i}(y_{i}))f_{i}(y_i){\rm d}y_1\cdots {\rm d}y_m.$$

When $b_{j}\in\dot{\Lambda}_{\beta_{j}}({{\mathbf{R}}}^{n})~(1\leq j\leq m)$, the authors establish the boundedness of $I_{\alpha,m}^{\vec{b}}$ on product Lebeasgue spaces, Triebel-Lizorkin spaces and Lipschitz spaces.

Key words: Multilinear fractional integral, Commutator, Triebel-Lizorkin space, Lipschitz function space

CLC Number:

42B25

MO Hui-Xia, ZHANG Zhi-Ying. Commutators Generated by Multilinear Fractional Integrals and Lipschitz Functions[J].Acta mathematica scientia,Series A, 2011, 31(5): 1447-1458.

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References

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Commutators Generated by Multilinear Fractional Integrals and Lipschitz Functions

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